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          Problem 109
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<h1 id="Problem-109"><a href="#Problem-109" class="headerlink" title="Problem 109"></a><a href="https://projecteuler.net/problem=109" target="_blank" rel="noopener">Problem 109</a></h1><hr>
<p><strong>Darts</strong></p>
<p>In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty.</p>
<center><img src="https://projecteuler.net/project/images/p109.gif" width="400" height="400" alt=""></center>

<p>The score of a dart is determined by the number of the region that the dart lands in. A dart landing outside the red/green outer ring scores zero. The black and cream regions inside this ring represent single scores. However, the red/green outer ring and middle ring score double and treble scores respectively.</p>
<p>At the centre of the board are two concentric circles called the bull region, or bulls-eye. The outer bull is worth 25 points and the inner bull is a double, worth 50 points.</p>
<p>There are many variations of rules but in the most popular game the players will begin with a score 301 or 501 and the first player to reduce their running total to zero is a winner. However, it is normal to play a “doubles out” system, which means that the player must land a double (including the double bulls-eye at the centre of the board) on their final dart to win; any other dart that would reduce their running total to one or lower means the score for that set of three darts is “bust”.</p>
<p>When a player is able to finish on their current score it is called a “checkout” and the highest checkout is 170: T20 T20 D25 (two treble 20s and double bull).</p>
<p>There are exactly eleven distinct ways to checkout on a score of 6:</p>
<table>
<thead>
<tr>
<th align="center">&nbsp;</th>
<th align="center">&nbsp;</th>
<th align="center">&nbsp;</th>
</tr>
</thead>
<tbody><tr>
<td align="center">D3</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">D1</td>
<td align="center">D2</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">S2</td>
<td align="center">D2</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">D2</td>
<td align="center">D1</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">S4</td>
<td align="center">D1</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">S1</td>
<td align="center">S1</td>
<td align="center">D2</td>
</tr>
<tr>
<td align="center">S1</td>
<td align="center">T1</td>
<td align="center">D1</td>
</tr>
<tr>
<td align="center">S1</td>
<td align="center">S3</td>
<td align="center">D1</td>
</tr>
<tr>
<td align="center">D1</td>
<td align="center">D1</td>
<td align="center">D1</td>
</tr>
<tr>
<td align="center">D1</td>
<td align="center">S2</td>
<td align="center">D1</td>
</tr>
<tr>
<td align="center">S2</td>
<td align="center">S2</td>
<td align="center">D1</td>
</tr>
</tbody></table>
<p>Note that D1 D2 is considered <strong>different</strong> to D2 D1 as they finish on different doubles. However, the combination S1 T1 D1 is considered the <strong>same</strong> as T1 S1 D1.</p>
<p>In addition we shall not include misses in considering combinations; for example, D3 is the <strong>same</strong> as 0 D3 and 0 0 D3.</p>
<p>Incredibly there are 42336 distinct ways of checking out in total.</p>
<p>How many distinct ways can a player checkout with a score less than 100?</p>
<hr>
<p><strong>飞镖</strong></p>
<p>在飞镖游戏中，玩家需向靶子上投掷三枚飞镖；靶子被分成了二十个相等面积的区域，并分别标上1至20。</p>
<center><img src="https://projecteuler.net/project/images/p109.gif" width="400" height="400" alt=""></center>

<p>每一枚飞镖的分数还取决于它的位置。落在外围的红/绿色圈以外时为零分，落在黑/白色区域时为一倍得分，而落在外围和中间的红/绿色圈时分别为两倍和三倍得分。</p>
<p>在把子的正中心有两个同心圆，被称为靶心。射中靶心外圈得25分，射中靶心内圈则得双倍50分。</p>
<p>飞镖的规则有许多变种，但最热门的一种是，每个玩家从301分或501分开始，轮流投掷飞镖并减去得分，首先将自己的分数减少到恰好为0的玩家获胜。不过，通常会采用“双倍结束”规则，即玩家在最后一镖必须射中一个双倍区域（包括双倍的靶心内圈）才能判定获胜。若这一轮的得分使得玩家的分数减少到1分或更少，但最后一镖未射中双倍区域，则这一轮的得分“作废”。</p>
<p>玩家在目前的分数下能够获胜则被称为“结分”。最高的结分为170：T20 T20 D25（两个三倍20分和一个双倍靶心）。</p>
<p>当玩家分数为6时，恰好有11种结分方式：</p>
<table>
<thead>
<tr>
<th align="center">&nbsp;</th>
<th align="center">&nbsp;</th>
<th align="center">&nbsp;</th>
</tr>
</thead>
<tbody><tr>
<td align="center">D3</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">D1</td>
<td align="center">D2</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">S2</td>
<td align="center">D2</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">D2</td>
<td align="center">D1</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">S4</td>
<td align="center">D1</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center">S1</td>
<td align="center">S1</td>
<td align="center">D2</td>
</tr>
<tr>
<td align="center">S1</td>
<td align="center">T1</td>
<td align="center">D1</td>
</tr>
<tr>
<td align="center">S1</td>
<td align="center">S3</td>
<td align="center">D1</td>
</tr>
<tr>
<td align="center">D1</td>
<td align="center">D1</td>
<td align="center">D1</td>
</tr>
<tr>
<td align="center">D1</td>
<td align="center">S2</td>
<td align="center">D1</td>
</tr>
<tr>
<td align="center">S2</td>
<td align="center">S2</td>
<td align="center">D1</td>
</tr>
</tbody></table>
<p>注意D1 D2被认为是<strong>不同于</strong>D2 D1的结分方式，因为它们最后的双倍不同。不过，组合S1 T1 D1和T1 S1 D1就被认为是<strong>相同的</strong>结分方式。</p>
<p>另外，我们在计算组合时，我们不考虑脱靶的情况；例如，D3和0 D3以及0 0 D3就是<strong>相同的</strong>结分方式。</p>
<p>令人惊奇的是一共有42336种不同的结分方式。</p>
<p>当玩家分数小于100时，一共有多少种不同的结分方式？</p>
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